Scientific Method —

Why photons are always neither here nor there

A nice piece of theoretical physics puts limits on how small a space a photon …

Photons and electrons are sort of the bread and butter of quantum physics. In introductory courses, a historical introduction will typically start by exploring the emission of light from a hot object and continue on to examining why, depending on the substance, light can cause the emission of electrons. The point of this is to describe how light comes in discrete chunks, called photons, while electrons, already known to be discrete, were later discovered to have wave-like properties as well.

But, despite this early introduction, the photon is a bit of a problem, because it has turned out to be impossible to identify a position operator for it. Photons can be described by creation and annihilation operators that are used to destroy one photon at one location and create a new photon at a different location. However, the full existence of the photon isn't described simply by creation and destruction—its movement between the two locations shouldn't be ignored. Now, a pair of Polish researchers show why the position operator is problematic.

To put this in perspective, tracking the humble electron in space is relatively simple. A position operator exists, and we can just repeatedly apply it and test it against measurements. Of course, a measurement can reveal the electron's position to arbitrary precision—provided we don't want to know anything about its momentum. In light of this, researchers expected to be able to find an equivalent position operator for the photon and, in principle, perform the same experiments. Unfortunately, this has not proved to be simple.

The key reason for this is that a photon is constructed from coupled electric and magnetic fields. The oscillation of these fields is what drives the photon through space, so a position operator actually needs to be described in terms of both fields. This latest research, published in Physical Review A, examines the behavior of these two fields at the single photon level.

To perform their analysis, the authors had to use a rather unusual formulation of electromagnetism. This alternative formulation is valid and has the advantage of not needing to construct photons from an infinite summation of mathematical objects—none of which have a definite position in space, either. Using this formulation, the researchers show that it is possible to localize either the electric or the magnetic field of the photon to arbitrary precision, although you can't do both at the same time.

It gets worse though, because localized states for a single field simply don't stay that way. Instead, as time passes, the localized field explodes outwards, while the delocalized field contracts. These findings tell us two things: first, unlike electrons, photons really can't be localized to an arbitrary precision, and second, a position operator is meaningless because there really is no position to operate on.

I don't think anyone in the theoretical physics community will be hugely surprised by this result. Furthermore, it probably doesn't mean a lot in terms of current work in the field of quantum optics because researchers there have learned to use creation and annihilation operators anyway. What this work does is highlight why there were no other options.

Chris Lee
Chris writes for Ars Technica's science section. A physicist by day and science writer by night, he specializes in quantum physics and optics. He lives and works in Eindhoven, the Netherlands. Emailchris.lee@arstechnica.com//Twitter@exMamaku

19 Reader Comments

Isn't a bit like trying to localize an ocean wave? You can identify when the wave crest will strike a bulkhead at a specific location. But, what does it mean to talk about the location of the wave? The energy of the wave is not concentrated at one location, but is distributed throughout the wave and is constantly interacting with other waves. The ocean wave at least has a medium so is easier to conceptualize.

I knew the pic looked familiar ... that's the photon shutter at beamline 9ID at the Advanced Photon Source. I guess you guys must have pulled it from our website - I'm honored!! However, I'm afraid Undulator x-ray radiation does not have much to do with the topic discussed here; it is more like >10^12 s^-1 photons of >keV energies.

I'm not a physicist nor a mathematician so I'm allowed to ask a stupid question.

I get that a photon is first an electric charge and then a magnetic charge, oscillating between the two.

I get that the photon has no fixed size, and thus any "position" below a certain precision is pretty hard to pin down because of the size changes in the photon.

However a photon *does* have a position, oscillation or no. I didn't understand the explanation the article offered as to why it couldn't be calculated.

Next stupid question, what exactly is a photon made of? If it has both an electric and magnetic charge then why isn't it an electron? What exactly is it made of? Finally, is it safe to say a photon has component parts, as it seems it must? *Something* is oscillating between electric and magnetic charge, what is that something?

I wasn't aware that this was a problem, and I'm not convinced that they've found the solution. Here's the hole: throughout the paper they repeatedly point out that they do not make use of Maxwell's equations. I'm not certain of the technical reason why this is necessary, but I assume that it is critical to their argument that they do not otherwise why dwell on it so frequently. In equation 8 they define the projection operators for the helicity operator, but they only do so for "divergence-free vectors [sic]." The electric and magnetic fields are only divergence free in a region if two conditions are met:

they obey Maxwell's equations, and

there is no charge in the region.

Since the proof that is the crux of their argument (found in Appendix B) depends entirely on this definition of the helicity projection operator, the argument fails. They would need to either generalize the helicity operator to apply to fields that have a divergence and then further generalize their argument, or find a way to relax the condition they took great pains to claim they satisfied and then failed to.

Originally posted by Wolf One Net:I'm not a physicist nor a mathematician so I'm allowed to ask a stupid question.

I get that a photon is first an electric charge and then a magnetic charge, oscillating between the two.

Not quite. First, nobody has ever seen a magnetic charge, and I personally think it unlikely that we ever will. Next, photons aren't oscillating charges, they're oscillations in the fields. To illustrate the difference, consider a string on a violin. When you pluck the violin string and release it, or draw a bow across it, the string vibrates. You can find good descriptions of how the dynamics of a vibrating string works just about anywhere. The analogy is that the sound waves on the string are the photons, the string represents the electric and magnetic fields, and the force the finger exerts on the string is like charge. The details of the mathematics are a little more challenging, of course, and those complications do have some important consequences, but the intuition is the same.

quote:

I get that the photon has no fixed size, and thus any "position" below a certain precision is pretty hard to pin down because of the size changes in the photon.

However a photon *does* have a position, oscillation or no. I didn't understand the explanation the article offered as to why it couldn't be calculated.

You are quite correct. You can define a position of a wave pulse on a string, for instance, by summing the square of the displacement of the string times the position of each displacement and then dividing by the sum of the square displacement. I wasn't aware that there was a problem with this for photons, and I'm too lazy to dig through the literature for the specifics of the debate. I do know, however, that that's exactly how we define the position of electron waves.

quote:

Next stupid question, what exactly is a photon made of? If it has both an electric and magnetic charge then why isn't it an electron? What exactly is it made of? Finally, is it safe to say a photon has component parts, as it seems it must? *Something* is oscillating between electric and magnetic charge, what is that something?

Good question. The answer is: nobody knows what the 'string' is. People used to think that since light was a wave there had to be some [em]thing[/em] that was vibrating (like the violin string), and thus the search for luminiferous ether. They didn't find it. Best we can say is: the electric and magnetic fields exist everywhere, they satisfy certain properties, and when a jiggle passes through them we call it a photon.

As for what's oscillating between the electric and magnetic fields, the answer is simple: energy. That's the fundamental take-home about waves: they're energy traveling through something else.

quote:

Lastly, why would the oscillation cause movement through space?

Told you they were stupid questions.

Well, it doesn't have to. You can make a standing wave even with photons. For instance, in a microwave cavity you can set up standing oscillations in the electric and magnetic fields just like with the violin string. In this case the walls of the cavity serve the same purpose as the bridge and top nut on a violin.

Originally posted by s0k0tr0k0:I knew the pic looked familiar ... that's the photon shutter at beamline 9ID at the Advanced Photon Source. I guess you guys must have pulled it from our website - I'm honored!! However, I'm afraid Undulator x-ray radiation does not have much to do with the topic discussed here; it is more like >10^12 s^-1 photons of >keV energies.

I'm glad you're honored.

It's really hard to find illustrative images (especially for what's an entirely theoretical article, in this case), and the journals and authors don't generally respond quickly enough for our publishing schedule, so we have to settle for images that are evocative, in many cases. .gov sites are exceptionally good in this regard, since the images aren't copyrighted.

Originally posted by Wolf One Net:However a photon *does* have a position

Sort of? I really don't know what I'm talking about, but AFAIK, you can't use a single three dimensional coordinate to define where a single photon is at a given point in time. We can identify a single point of impact, but before that the photon seems to simultaneously occupy a whole range of space, which influences the probability of the point of impact.

Look at the double-slit experiment and how various photons interfere with one another to produce a diffraction pattern. Now, try to wrap your head around the fact that the same pattern is produced when you output a single photon at a time. A photon's wave will actually interfere with itself.

Looking again at the paper and the definition of the helicity operator given, it's actually pretty easy to generalize the projection operators. Just multiply them by chi^2 - on the solenoidal component of a vector field, V_s, chi^2 V_s = V_s, and for the irrotational component chi V_i = 0. The full set of projection operators are then: (chi^2 +/- chi)/2, and (1 - chi^2). His formulation of d and b being equivalent to D and B requires that (1 - chi^2) on D and B be zero, thus that they obey Maxwell's equations.

So they can't just generalize the projection operator - I just did, and it didn't help them. They need to address why, "we didn't use Maxwell's equations," is important enough to mention so much.

Even if they say, "Ok, fine, we'll specialize to the case of Maxwell's equations holding," then I'd like to hear about the case with inhomogeneous Maxwell's equations. See, if you specialize to the case where Maxwell's equations hold then they have only proven that you cannot localize a photon in the absence of any charges/currents. And they can't use the excuse that electrons etc are point like and thus irrelevant almost everywhere, either. If you look at the Lagrangian we use to define this QED it has the form: integral( free Maxwell + free electrons -ie A^\mu bar(Psi) \gamma^\mu Psi), where Psi is the electron field strength (this can be related to electron wave functions). In other words, if you have a non-localized electron wave function then calculating with point-like electrons is an approximation.

Originally posted by s0k0tr0k0:I knew the pic looked familiar ... that's the photon shutter at beamline 9ID at the Advanced Photon Source. I guess you guys must have pulled it from our website - I'm honored!! However, I'm afraid Undulator x-ray radiation does not have much to do with the topic discussed here; it is more like >10^12 s^-1 photons of >keV energies.

I'm glad you're honored.

It's really hard to find illustrative images (especially for what's an entirely theoretical article, in this case), and the journals and authors don't generally respond quickly enough for our publishing schedule, so we have to settle for images that are evocative, in many cases. .gov sites are exceptionally good in this regard, since the images aren't copyrighted.

Good point. In my previous response I was caught up in the technical "position operator," which is a fancy way of saying that you can find an average position somewhat akin to the center of mass. The position in that sense is, of course, not a complete description of a real photon the way the position of a point particle or even a billiard ball is.

"Why Classical Mechanics limited"sites.google.com/site/socialcapital1/Home In this article, from the perspective of classical philosophy, I explain why neither Newton's laws nor the laws of Coulomb’s interaction of charges work in quantum mechanics. Then, from the same point of view, I explain the nature of the conflicting definitions of micro particles etc...

It's been a long time, but I always thought that the lack of localization was just the result that the QED Hamiltonian integrals were non-convergent. ie. do we really have a photon or do we have a veritable zoo of particles and quanta spreading out in space and time?

Using this formulation, the researchers show that it is possible to localize either the electric or the magnetic field of the photon to arbitrary precision, although you can't do both at the same time.

It's kindof Heisenberg-ish? Instead of position and momentum, or time and energy, it's electric and magnetic?

This doesn't make sense to me, because I was under the impression that the "disturbance" that's propagating through the electric and magnetic field was coupled in some way; i.e. if you know where one is, the other is nearby and orthogonal.

So...if you can't nail down the position of a photon, then after it is created, it exists everywhere along the path on the way to its destination? And this allows the probability of it getting to a particular destination to be affected by all the paths it could take interfering with each other (i.e. in the double-slit, a single electron interferes with itself, right?)

I'm seeing in my head a sort of tentacle-like whiskers or feelers that are reaching out from the photon source in search of a photon sink, and once a sink is found all the energy sortof collapses on the sink.

Using this formulation, the researchers show that it is possible to localize either the electric or the magnetic field of the photon to arbitrary precision, although you can't do both at the same time.

It's kindof Heisenberg-ish? Instead of position and momentum, or time and energy, it's electric and magnetic?

This doesn't make sense to me, because I was under the impression that the "disturbance" that's propagating through the electric and magnetic field was coupled in some way; i.e. if you know where one is, the other is nearby and orthogonal.

Yep. See, in quantum field theory the fundamental operators aren't position and momentum. The fundamental operators are the field strength and its canonically conjugate momentum (the time derivative of the hermitian conjugate for complex scalar fields, time derivative for real scalars, etc). You can then build the Heisenberg type position and momentum operators out of integrals of the field strength operators. In the case of E&M, the non-commuting variables are E and B (or you could use the vector potential A and the electric field E, but then you run into tricky questions about how to ensure gauge invariance).

Bottom line, here's what they did: using the commutation relations that make quantum mechanics what it is and half of Maxwell's equations in a space without charge, they proved that you cannot localize a photon. Where you're getting confused is that the "where E is B is" bit comes from the half of Maxwell's equations that they didn't use, because those are the equations that relate the dynamics of the electric and magnetic field.

quote:

So...if you can't nail down the position of a photon, then after it is created, it exists everywhere along the path on the way to its destination? And this allows the probability of it getting to a particular destination to be affected by all the paths it could take interfering with each other (i.e. in the double-slit, a single electron interferes with itself, right?)

No, they screwed up because in order to create a photon you need to have a charge hanging around, which would spoil a key step in their proof. The other thing to keep in mind is that even their de-localized photons can be exponentially damped. That may not seem like much, but when you realize that the electrons in a bound orbital around an atom are only exponentially damped, too, it could make for a pretty convenient match between how delocalized the electron is and how delocalized the photon it produces in a transition is.